Using the derivative of f(x) given below, determine the critical points of f(x).
-$$f ^ { \prime } ( x ) = ( x - 5 ) e ^ { - x }$$
A) -6
B) -5
C) 5
D) 6

Question

Quizplus · Accepted Answer

To find the critical points of a function, we need to find the values of x where the derivative is equal to zero or undefined.
Setting the derivative equal to zero, we get:
$$0=(x-5)e^{-x}$$
This is true when either factor is equal to zero, so we have:
$$x-5=0  \quad \text{or} \quad e^{-x}=0$$
The second equation has no solutions since exponential functions are always positive. Thus, we only need to solve the first equation to find the critical point:
$$x-5=0$$
$$x=5$$
Therefore, the only critical point is x=5, which is choice C.

Using the Derivative of F(x) Given Below, Determine the Critical f′(x)=(x−5)e−xf ^ { \prime } ( x ) = ( x - 5 ) e ^ { - x }f′(x)=(x−5)e−x

Using the Derivative of F(x) Given Below, Determine the Critical $f ^ { \prime } ( x ) = ( x - 5 ) e ^ { - x }$